HYPERBOLICITY AND BIFURCATIONS IN HOLOMORPHIC FAMILIES OF POLYNOMIAL SKEW PRODUCTS

We study holomorphic families of polynomial skew products, i.e., polynomial endomor-phisms of C(2 )of the form F(z,w) = (p(z),q(z,w)) that extend to holomorphic endomorphisms of P-2(C). We prove that stability in the sense of [Berteloot, Bianchi, and Dupont, 2018] preserves hy-perbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family z(2) + c. We also precisely describe the geometry of the bifurcation locus and current near the boundary of the parameter space. One of our tools is an asymptotic equidistribution property for the bifurcation cur-rent. This is established in the general setting of families of endomorphisms of P-k, and is the first equidistribution result of this kind for holomorphic dynamical systems in dimension larger than one.